Warm Up Exercises

1. Write a matrix down on a piece of paper.  By moving the paper only, can you find the transpose of the matrix?

2. Epistemology of Transposes:

a. If you take the transpose twice, should you have?

b. Prove that no matrix can have the property that taking the transpose twice doesn't give the matrix itself.  As a corollary, argue limitations on any all-powerful being, and discuss the implications of such a limitation on the struggle between good and evil, freedom and tyranny, Yankees and Red Sox, and other important religious ideas.

c. If someone takes a transpose of a symmetric matrix in the forest, did it make a noise?

3. Write a general 4x4 matrix in terms of the Dirac gamma matricies and their bilinears.  Prove, by the linearity of the transposition operation and direct transposition of those objects, that any 4x4 matrix may be transposed.